It looks like you'll need to leave a single PC connected directly to the modem for a day or so. It'll be a pain, but that's exactly what TW will ask if you call them.
If you can confirm that the problem happens when the modem is connected directly to the PC (even better, let it happen on one PC then see if if happens on another - but that will take longer
) then I would have thought that should be enough to get them to send a tech out (eventually!)